Control engineering and finance (lecture notes in control and information sciences)

Lecture Notes in Control and Information Sciences 467

Selim S. Hacısalihzade

Control
Engineering
and Finance
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Lecture Notes in Control and Information
Sciences
Volume 467

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Selim S. Hacısalihzade

Control Engineering
and Finance

123


Selim S. Hacısalihzade
Department of Electrical and Electronics
Engineering
Boğaziçi University
Bebek, Istanbul
Turkey

ISSN 0170-8643
ISSN 1610-7411 (electronic)
Lecture Notes in Control and Information Sciences
ISBN 978-3-319-64491-2
ISBN 978-3-319-64492-9 (eBook)
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Acknowledgements

There is a long list of people to acknowledge and thank for their support in
preparing this book. I want to begin by thanking Jürg Tödtli who supported my
interest in the field of quantitative finance—even though I did not know the term at
that time—while we were at the Institute of Automatic Control at the ETH Zurich
many decades ago. This interest was triggered and then re-triggered in countless
discussions over the years with my uncle Ergün Yüksel. Special thanks are certainly
due to Manfred Morari, the former head of the Institute of Automatic Control at the
ETH Zurich, who encouraged me to write this book and to publish it in the Lecture
Notes in Control and Information Science series of Springer Verlag and who also
offered me infrastructure and library access during the preparation of the
manuscript.
Very special thanks go to Florian Herzog of Swissquant who supported me
while I was writing this book by offering the use of his lecture notes of Stochastic
Control, a graduate course he held at the ETH Zurich. I did so with gratitude in
Chapters 5 and 9. The data for the empirical study reported in Chapter 8 were
graciously supplied by Özgür Tanrverdi of Access Turkey Opportunities Fund for
several years, to whom I am indebted. I also want to thank Jens Galschiøt, the
famous Danish sculptor for allowing me to use a photograph of his impressive and
inspiring sculpture “Survival of the Fattest” to illustrate the inequality in global
wealth distribution.
Parts of this book evolved from a graduate class I gave at Boğaziçi University in
Istanbul during the last years and from project work by many students there,
notably Efe Doğan Yılmaz, Ufuk Uyan, Ceren Sevinç, Mehmet Hilmi Elihoş,
Yusuf Koçyiğit and Yasin Çotur. I am most grateful to Yasin, a Ph.D. candidate at
Imperial College in London now, who helped with calculations and with valuable
feedback on earlier versions of the manuscript.
I am indebted to my former student Yaşar Baytın, my old friend Sedat Ölçer,
Head of Computer Science and Engineering at Bilgi University in Istanbul, Bülent

Sankur, Professor Emeritus of Electrical and Electronics Engineering at Boğaziçi

vii

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viii

Acknowledgements

University, and especially my dear wife Hande Hacısalihzade for proofreading parts
of the manuscript and their most valuable suggestions.
I am grateful to Petra Jantzen and Shahid S. Mohammed at Springer for their
assistance with the printing of this volume.
Hande, of course, also deserves special thanks for inspiring me (and certainly not
only for writing limericks!), hours of lively discussions, her encouragement, and her
endless support during the preparation of this book.
Selim S. Hacısalihzade
Istanbul 2017


Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Control Engineering and Finance . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Probability and Stochastic Processes . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 History and Kolmogorov’s Axioms . . . . . . . . . . . . . . . . . . .
3.3 Random Variables and Probability Distributions. . . . . . . . . .
3.3.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Probability Distribution of a Discrete Random
Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . .

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2 Modeling and Identification . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 What Is a Model? . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Modeling Process . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Stock Prices . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Lessons Learned . . . . . . . . . . . . . . . . . .
2.4 Parameter Identification . . . . . . . . . . . . . . . . . . .
2.5 Mathematics of Parameter Identification . . . . . . .
2.5.1 Basics of Extremes . . . . . . . . . . . . . . . .
2.5.2 Optimization with Constraints . . . . . . . .
2.6 Numerical Methods for Parameter Identification .
2.6.1 Golden Section . . . . . . . . . . . . . . . . . . .
2.6.2 Successive Parameter Optimization . . . .
2.7 Model Validation . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

3.3.5

Multidimensional Distribution Functions
and Independence . . . . . . . . . . . . . . . . . . . . .
3.3.6 Expected Value and Further Moments . . . . . .
3.3.7 Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.8 Normal Distribution . . . . . . . . . . . . . . . . . . . .
3.3.9 Central Limit Theorem . . . . . . . . . . . . . . . . .
3.3.10 Log-Normal Distribution . . . . . . . . . . . . . . . .
3.4 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Mathematical Description of Stochastic Processes
with Distribution Functions . . . . . . . . . . . . . . . . . . . .
3.6 Stationary and Ergodic Processes . . . . . . . . . . . . . . . .
3.7 Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Some Special Processes . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Normal (Gaussian) Process . . . . . . . . . . . . . .

3.8.2 Markov Process . . . . . . . . . . . . . . . . . . . . . . .
3.8.3 Process with Independent Increments . . . . . .
3.8.4 Wiener Process . . . . . . . . . . . . . . . . . . . . . . .
3.8.5 Gaussian White Noise . . . . . . . . . . . . . . . . . .
3.9 Analysis of Stochastic Processes. . . . . . . . . . . . . . . . .
3.9.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . .
3.9.4 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.5 Brief Summary . . . . . . . . . . . . . . . . . . . . . . .
3.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Calculus of Variations . . . . . . . . . . . . . . . . . . . .
4.2.1 Subject Matter . . . . . . . . . . . . . . . . . . . .
4.2.2 Fixed Endpoint Problem . . . . . . . . . . . .
4.2.3 Variable Endpoint Problem . . . . . . . . . .
4.2.4 Variation Problem with Constraints . . . .
4.3 Optimal Dynamic Systems . . . . . . . . . . . . . . . . .
4.3.1 Fixed Endpoint Problem . . . . . . . . . . . .
4.3.2 Variable Endpoint Problem . . . . . . . . . .
4.3.3 Generalized Objective Function . . . . . . .
4.4 Optimization with Limited Control Variables . . .
4.5 Optimal Closed-Loop Control . . . . . . . . . . . . . . .
4.6 A Simple Cash Balance Problem . . . . . . . . . . . .
4.7 Optimal Control of Linear Systems . . . . . . . . . .
4.7.1 Riccati Equation . . . . . . . . . . . . . . . . . .
4.7.2 Optimal Control When Not All States
Are Measurable . . . . . . . . . . . . . . . . . . .

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Contents

xi

4.8
4.9

Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9.1 Remembering the Basics . . . . . . . . . . . . . . . . . . . . .
4.9.2 Time Optimization. . . . . . . . . . . . . . . . . . . . . . . . . .
4.9.3 Optimization with a Quadratic Objective Function .
4.10 Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.2 Static Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.3 Zero-Sum Games . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.4 The Co-Co Solution . . . . . . . . . . . . . . . . . . . . . . . .

4.10.5 Dynamic Games . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Stochastic Differential Equations . . . . . . . . . . . . . . . .
5.4 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Properties of Itô Integrals . . . . . . . . . . . . . . . . . . . . . .
5.6 Itô Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Solving Stochastic Differential Equations . . . . . . . . . .
5.7.1 Linear Scalar SDE’s . . . . . . . . . . . . . . . . . . .
5.7.2 Vector-Valued Linear SDE’s . . . . . . . . . . . . .
5.7.3 Non-linear SDE’s and Pricing Models . . . . . .
5.8 Partial Differential Equations and SDE’s . . . . . . . . . .
5.9 Solutions of Stochastic Differential Equations . . . . . .
5.9.1 Analytical Solutions of SDE’s . . . . . . . . . . . .
5.9.2 Numerical Solution of SDE’s . . . . . . . . . . . .
5.9.3 Solutions of SDE’s as Diffusion Processes . .
5.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Financial Markets and Instruments . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Time Value of Money . . . . . . . . . . . . . . . .
6.3 Financial Investment Instruments . . . . . . . .

6.3.1 Fixed Income Investments . . . . . . .
6.3.2 Common Stocks . . . . . . . . . . . . . .
6.3.3 Funds . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Commodities . . . . . . . . . . . . . . . . .
6.3.5 Forex . . . . . . . . . . . . . . . . . . . . . . .
6.3.6 Derivative/Structured Products. . . .
6.3.7 Real Estate . . . . . . . . . . . . . . . . . .
6.3.8 Other Instruments . . . . . . . . . . . . .
6.4 Return and Risk . . . . . . . . . . . . . . . . . . . . .

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xii

Contents

6.5

6.6
6.7
6.8

Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Utility Functions . . . . . . . . . . . . . . . . . . . . .
Role of the Banks in Capitalist Economies .
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . .

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193
195
197
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7 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Bond Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Bond Returns . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Bond Valuation . . . . . . . . . . . . . . . . . . . . . . . . .

7.6 Fundamental Determinants of Interest Rates
and Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Rating Agencies . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 The Yield Curve . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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207
209
209
211
213

8 Portfolio Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Measuring Risk and Return of Investments . . . . . . . .
8.3 Modern Portfolio Theory . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Measuring Expected Return and Risk
for a Portfolio of Assets . . . . . . . . . . . . . . . .
8.3.2 Efficient Frontier . . . . . . . . . . . . . . . . . . . . . .
8.4 Portfolio Optimization as a Mathematical Problem . . .
8.5 Portfolio Optimization as a Practical Problem. . . . . . .
8.6 Empirical Examples of Portfolio Optimization
Using Historical Data . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Bond Portfolios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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218
220

221
225

.........
.........
.........

228
233
239

9 Derivatives and Structured Financial Instruments . .
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . .
9.3 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Some Properties of Options . . . . . . . . . .
9.4.3 Economics of Options . . . . . . . . . . . . . .
9.4.4 Black-Scholes Equation . . . . . . . . . . . . .
9.4.5 General Option Pricing . . . . . . . . . . . . .

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Contents

xiii

9.5
9.6
9.7

Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Structured Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Appendix A: Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Appendix B: Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Appendix C: Normal Distribution Tables . . . . . . . . . . . . . . . . . . . . . . . . . 287
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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Chapter 1

Introduction

What we need is some financial engineers.
— Henry Ford
Diversifying sufficiently among uncorrelated risks can reduce
portfolio risk toward zero. But financial engineers should know
that’s not true of a portfolio of correlated risks.
— Harry Markowitz

1.1 Control Engineering and Finance
At first glance, the disciplines of finance and control engineering may look as unrelated as any two disciplines could be. However, this is true only to the uninitiated
observer. For trained control engineers, the similarities of the underlying problems
are striking. One, if not the main, intent of control engineering is to control a process
in such a way that it behaves in the desired manner in spite of unforeseen disturbances acting upon it. Finance, on the other hand, is the study of the management of
funds with the objective of increasing them, in spite of unexpected economical and
political events. Once formulated this way, the similarity of control engineering to
finance becomes obvious.
Perhaps the most powerful tool for reducing the risk of investments is diversification. If one can identify the risks specific to a country, a currency, an instrument
class and an individual instrument, the risk conscious investor—as they should all
be—ought to distribute her wealth among several countries, several currencies and
among different instrument classes like, for instance, real estate, bonds of different
issuers, equity of several companies and precious metals like gold.

© Springer International Publishing AG 2018
S. S. Hacısalihzade, Control Engineering and Finance, Lecture Notes in Control

and Information Sciences 467, />
1


2

1 Introduction

Clearly, within equity investments, it makes sense to invest in several companies,
ideally in different countries. In this context, as discussed in detail in Chapter 8,
Modern Portfolio Theory attempts to maximize the expected return of a portfolio for
a given amount of risk, or equivalently minimize the amount of risk for a given level
of expected return by optimizing the proportions of different assets in the portfolio.
Chapters 4, 5, and 8 show that Optimal Stochastic Control constitutes an excellent tool for constructing optimal portfolios. The use of financial models with control
engineering methods has become more widespread with the aim of getting better and
more accurate solutions. Since optimal control theory is able to deal with deterministic and stochastic models, finance problems can often be seen as a mixture of the
two worlds.
A generic feedback control system is shown in Figure 1.1. The system is composed
of two blocks, where P denotes the process to be controlled and C the controller. In
this representation r stands for the reference, e the error, u the control input, d the
disturbance, and y the output.
The structure in Figure 1.2 can be used as a theoretical control model for dynamic
portfolio management which suggests three stages: the Estimator estimates the return
and the risk of the current portfolio and its constituents, the Decider determines the
timing of the portfolio re-balancing by considering relevant criteria and the Actuator
changes the current portfolio to achieve a more desirable portfolio by solving an
optimization problem involving the model of the portfolio. In many cases, also due
to regulatory constraints, the Actuator involves a human being but it can compute
and execute buy/sell transactions without human involvement as well.
Looking at these two figures, one can observe that they have a similar structure in

the sense of a feedback loop. Therefore, finance or investing, and more specifically
the problem of portfolio management can be regarded as a control problem where r
is the expected return of the portfolio, e is the difference between the expected and
Fig. 1.1 A generic feedback
control system

Fig. 1.2 A theoretical
control model for dynamic
portfolio management

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1.1 Control Engineering and Finance

3

the actual portfolio returns, P is the portfolio, C is the algorithm which maximizes
the portfolio return under certain constraints and u are the buy/sell instructions to
re-balance the portfolio.

1.2 Outline
Numerous books have been published about the solution of various financial problems using control engineering techniques as researchers conversant in both fields
became aware of their similarities. Many such specific examples can be found in the
literature, especially in the areas of optimal control and more recently in stochastic
control. Therefore, even a superficial review of this particular interdisciplinary field
covering all subjects would have to weigh several volumes. Hence, any textbook
in this field has either to take one specific application of control theory in finance
and explore it in depth or be an eclectic collection of several problems. This Book
chooses to take the latter path also because several excellent textbooks of the former

type are already available.
This Volume presents a number of different control engineering applications in
finance. It is intended for senior undergraduate or graduate students in electrical engineering, mechanical engineering, control engineering, industrial engineering and
financial engineering programs. For electrical/mechanical/control/industrial engineering students, it shows the application of various techniques they have already
learned in theoretical lectures in the financial arena. For financial engineering students, it shows solutions to various problems in their field using methods commonly
used by control engineers. This Book should also appeal to students and practitioners
of finance who want to enhance their quantitative understanding of the subject.
There are no sine qua non prerequisites for reading, enjoying and learning from
this textbook other than basic engineering mathematics and a basic understanding
of control engineering concepts. Nevertheless, the first half of the Book can be seen
as a refresher of or an introduction to several tools like mathematical modeling of
dynamic systems, analysis of stochastic processes, calculus of variations and stochastic calculus which are then applied to the solution of some financial problems in the
second part of the Book. It is important to remember that this is by no means a mathematics book even though it makes use of some advanced mathematical concepts.
Any definitions of these concepts or any derivations are neither rigorous nor claim
to be complete. Therefore, where appropriate, the reader looking for mathematical
precision is referred to standard works of mathematics.
After this introductory Chapter, the Book begins by discussing a very important
topic which is often neglected in most engineering curricula, namely, mathematical
modeling of physical systems and processes. Chapter 2 discusses what a model is
and gives various classification methods for different types of models. The actual
modeling process is illustrated using the popular inverted pendulum and the stock
prices. The generic modeling process is explained, highlighting parameter identification using various numerical optimization algorithms, experiment design, and model


4

1 Introduction

validation. A small philosophical excursion in this Chapter is intended to alert the
reader to the differences between reality and models.

Chapter 3, Probability and Stochastic Processes, begins by introducing Kolmogorov’s axioms and reviewing some basic definitions and concepts of random
variables such as moments, probability distributions, multidimensional distribution
functions and statistical independence. Binomial and normal distributions are discussed and related to each other through the central limit theorem. Mathematical
description and analysis of stochastic processes with emphasis on several relevant
special classes like stationary and ergodic processes follows. Some special processes
such as the normal process, the Wiener process, the Markov process and white noise
are discussed. Students not comfortable with the concepts in this Chapter should
study them very carefully, since they constitute an important mathematical foundation for the rest of the Book.
Chapter 4, Optimal Control, is thought as an introduction to this vast field. It begins
with calculus of variations and goes through the fixed and variable endpoint problems
as well as the variation problem with constraints. Application of these techniques
to dynamic systems leads to the solution of the optimal control problem using the
Hamilton-Jacobi method for closed-loop systems where all the states have to be fed
back to assure optimality independent of initial conditions. Pontryagin’s minimum
principle is discussed in connection with optimal control in the case of the control
variables being limited (as they always are in practice). Special emphasis is given
to optimal control of linear systems with a quadratic performance index leading to
the Riccati equation. Dynamic programming is briefly presented. The Final Section
of this Chapter gives an introduction to Differential Games, thus establishing a firm
link between Optimal Control and Finance.
Chapter 5, Stochastic Analysis, constitutes, perhaps, the heart of the Book. It
begins with a rigorous analysis of white noise and introduces stochastic differential
equations (SDE’s). Stochastic integration and Itô integrals are introduced and their
properties are scrutinized. Stochastic differentials are discussed and the Itô lemma
is derived. Methods for solving SDE’s using different techniques including Itô calculus and numerical techniques are shown for scalar and vector valued SDE’s both
in the linear and the non-linear cases. Several stochastic models used in financial
applications are illustrated. The connection between deterministic partial differential equations and SDE’s is pointed out.
Chapter 6, Financial Markets and Instruments is written in an informal and colloquial style, because it might be the first instance where an engineering student
encounters the capital markets and various financial instruments. The Chapter begins
with the concept of time value of money and introduces the main classes of financial

instruments. It covers the fixed income instruments like savings accounts, certificates
of deposit, and introduces Bonds. The Chapter then moves on to talk about a number
of other financial instruments, including common stocks, various types of funds and
derivative instruments. Fundamental concepts relating to risk, return and utility are
discussed in this Chapter. Finally, the role and importance of the banks for the proper
functioning of the economy are presented.

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1.2 Outline

5

Special emphasis is given to Bonds in Chapter 7. Bond returns and valuations are
derived heuristically based on the return concept defined in the previous Chapter. Fundamental determinants of interest rates and bond yields, together with the Macaulay
duration are discussed with examples. The yield curve is presented together with its
implications for the economy.
Chapter 8, Portfolio Management, is dedicated to the problem of maximizing the
return of a portfolio of stocks over time while minimizing its risk. The fundamental
concept of the Efficient Frontier is introduced within the context of this quintessential
control engineering problem. The problem of choosing the sampling frequency is
formulated as the question of choosing the frequency of re-balancing a stock portfolio
by selling and buying individual stocks. Empirical studies from the Swiss, Turkish
and American stock markets are presented to address this question. Additional algorithms for managing stock portfolios based on these empirical studies are proposed.
Management of bond portfolios using the Vašíˇcek model and static bond portfolio
management methods finish off this Chapter.
Chapter 9, Derivative Financial Instruments, begins by reviewing forward contracts, futures and margin accounts. Options are discussed in detail in this Chapter.
The Black-Scholes options pricing model is presented and the pricing equation is
derived. Selected special solutions of this celebrated equation for the pricing of European and American options are shown. Finally, the use of options in constructing

structured products to enhance returns and cap risks in investing is demonstrated.
There are three Appendices which contain various mathematical descriptions of
dynamic systems including the concept of state space; matrix algebra and matrix
calculus together with some commonly used formulas; and the requisite standardized
normal distribution tables.
Every Chapter after this one includes three types of exercises at its conclusion:
Type A exercises are mostly verbal and review the main points of the Chapter; they
aim to help the reader gauge her1 comprehension of the topic. Type B exercises
require some calculation and they are intended to deepen the understanding of the
methods discussed in the Chapter. Type C exercises involve open ended questions and
their contemplation typically requires significant time, effort and creativity. These
questions can qualify as term or graduation projects, and may indicate directions for
thesis work.
Five books are enthusiastically recommend, especially to those readers who have
an appetite for a less technical take on the workings of the financial markets. This
short reading list should accompany the reader during his perusal of this Book and
the odds are that he will return to these classics for many years to come.
• “The Physics of Wall Street: A Brief History of Predicting the Unpredictable” by
James Owen Weatherall, 2014,
• “The Drunkard’s Walk: How Randomness Rules Our Lives” by Leonard Mlodinow, 2008,
1 To

avoid clumsy constructs like “his/her”, where appropriate, both male and female personal
pronouns are used throughout the Book alternatingly and they are interchangeable with no preference
for or a prejudice against either gender.


6

1 Introduction

• “The Intelligent Investor” by Benjamin Graham,2 2006, [47]
• “The Black Swan: The Impact of the Highly Improbable” by Nassim Nicholas
Taleb, 2007, and
• “Derivatives: The Tools that Changed Finance” by the father and son Phelim and
Feidhlim Boyle, 2001.
It is humbling to see how inherently challenging subjects like randomness, investing, and financial derivatives are so masterfully explained in these books for laypersons in excellent prose.

2 Benjamin

Graham, British-American economist and investor (1894– 1976).

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Chapter 2

Modeling and Identification

“That’s another thing we’ve learned from your Nation,” said
Mein Herr, “map-making. But we’ve carried it much further
than you. What do you consider the largest map that would
be really useful?”
“About six inches to the mile.”
“Only six inches!” exclaimed Mein Herr. “We very soon got to six
yards to the mile. Then we tried a hundred yards to the mile.
And then came the grandest idea of all! We actually made a
map of the country, on the scale of a mile to the mile!”
“Have you used it much?” I enquired.
“It has never been spread out, yet,” said Mein Herr: “the farmers

objected: they said it would cover the whole country, and shut
out the sunlight! So we now use the country itself, as its own
map, and I assure you it does nearly as well.”
— Lewis Carroll, Sylvie and Bruno Concluded

2.1 Introduction
Most control engineering exercises begin with the phrase “Given is the system with
the transfer function ...”. Alas, in engineering practice, a mathematical description
of the plant which is to be controlled is seldom available. The plant first needs to be
modeled mathematically. In spite of this fact, control engineering courses generally
do not spend much time on modeling. Perhaps this is because modeling can be said
to be more of an art than a science. This Chapter begins by defining what is meant
by the words model and modeling. It then illustrates various types of models, studies
the process of modeling and concludes with the problem of parameter identification
and related optimization techniques.

© Springer International Publishing AG 2018
S. S. Hacısalihzade, Control Engineering and Finance, Lecture Notes in Control
and Information Sciences 467, />
7


8

2 Modeling and Identification

2.2 What Is a Model?
Indeed, what is a model? Probably one gets as many different responses as the number
of persons one poses this question.1 Therefore, it is not surprising that MerriamWebster Dictionary offers 23 different definitions under that one entry. The fourth
definition reads “a usually miniature representation of something” as in a “model

helicopter”. The ninth definition reads, rather unglamorously, “one who is employed
to display clothes or other merchandise” as in Heidi Klum, Naomi Campbell or
Adriana Lima depending on which decade you came off age. Well, if your interest
in models is limited to these definitions you can close this Book right now.
Definitions 11 and 12 read “a description or analogy used to help visualize something (as an atom) that cannot be directly observed” and “a system of postulates,
data and inferences presented as a mathematical description of an entity or state of
affairs; also: a computer simulation based on such a system <climate models>”.
These definitions are closer to the sense of the word model that is used throughout
this Book.
These definitions might be considered too general to be of any practical use. Let us
therefore think of a model as an approximate representation of reality. One obvious
fact, forgotten surprisingly often, is that a model is an abstraction and that any model
is, by necessity, an approximation of reality. Consequently, there is no one “true
model”, rather there are models which are better than others. But what does “better”
mean? This clearly depends on the context and the problem at hand.
Example: An illustrative example which can be found in many high school physics
books is dropping an object from the edge of a table and calculating the time it will
take for the object to hit the floor. Assuming no air friction, here one can write the
well-known Newtonian2 equation of motion which states that the object will move
with a constant acceleration caused by the weight of the object, which again is caused
by the Earth’s gravitational attraction:
m x(t)
¨ = mg .

(2.1)

x(t)
¨ denotes the second derivative of the distance x (acceleration) of the object
from the edge of the table as a function of time after the drop; g is the Earth’s
acceleration constant. m is the mass of the object but it is not relevant, because it can

be canceled away. Solving the differential Equation (2.1) with the initial conditions
x(0) = 0 and x(0)
˙
= 0 results in
x(t) =

1 2
gt .
2

1 This

(2.2)

Chapter is an extended version of Chapter 3 in [51].
Newton, English astronomer, physicist, mathematician (1642–1726); widely recognized as
one of the most influential scientists of all time and a key figure in the scientific revolution; famous
for developing infinitesimal calculus, classical mechanics, a theory of gravitation and a theory of
color.

2 Isaac

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2.2 What Is a Model?

9

Solving (2.2) for the impact time ti with the height of the table denoted as h results

in the well-known formula
ti =

2h
.
g

(2.3)

Let us now consider the problem of dropping an object attached to a parachute
from an aircraft. Here, one can no longer assume there is no air friction. The very
reason for deploying a parachute is to make use of the air friction to slow down the
impact velocity of the drop. Hence, to calculate the impact time with better accuracy,
(2.1) needs to be modified to account for the air friction, a force which increases with
the square of velocity:
x(t)
¨ = g − k S x˙ 2 (t) .

(2.4)

x(t)
˙ denotes the first derivative of the distance x (velocity) of the object from the
drop height, k is the viscous friction coefficient and S is the effective surface area
of the parachute. It is no longer possible to have a simple analytical solution of this
non-linear differential equation and one has to revert to numerical means to solve
it. When the terminal velocity is reached (i.e., k S x˙ 2 = g ⇒ x¨ = 0) the object stops
accelerating and keeps falling with a constant velocity.
This example shows that the same physical phenomenon can be modeled in different ways with varying degrees of complexity. It is not always possible to know as
in this case which model will give better results a priori. Therefore, one often speaks
of the “art” of modeling. An engineer’s approach to modeling might be to begin with

certain restrictive assumptions leading to a simple model. At further steps, these
assumptions can be relaxed to modify the initial simple model and to account for
further complications until a model is attained which is appropriate for the intended
purpose.
There are many different types of models employed in science and philosophy.
The reader might have heard of mental models, conceptual models, epistemological
models, statistical models, scientific models, economic models or business models
just to name a few. This Book limits itself to scientific and mathematical models.
Scientific modeling is the process of generating abstract, conceptual, graphical or
mathematical models using a number of methods, techniques and theories. The general purpose of a scientific model is to represent empirical phenomena in a logical
and objective way. The process of modeling is presented in the next section.
A mathematical models helps to describe a system using mathematical tools. The
use of mathematical models is certainly not limited to engineering or natural sciences
applications. Mathematical models are increasingly being used in social sciences like
economics, finance, psychology and sociology3,4 .
3 Isaac

Asimov, American writer (1920–1992).
science fiction enthusiast is unaffected by the Asimovian character Hari Seldon’s “psychohistory”, which combines history, sociology and statistics to make general predictions about the

4 Which


10

2 Modeling and Identification

Fig. 2.1 Prey and predator populations as a numerical solution of the set of equations (2.5). Different
shades of gray indicate different initial conditions [59]

Mathematical models can be classified under the following dichotomic headings:
Inductive versus deductive models: A deductive model is based on a physical theory.
In the example above, Newton’s laws of motion were used to model the movement of a
falling object. An inductive model, on the other hand, is based on empirical findings
and their generalizations without forming a generally applicable law of nature. A
well-known example is the set of Lotka-Volterra equations used in modeling the
dynamics of biological systems in which two species interact as predator and prey
[78]. Here the rate of change of the number of preys, x (say, rabbits) is proportional
to the number of preys who can find ample food and who can breed, well, like rabbits.
This exponential growth is corrected by prey-predator encounters (say, with foxes).
Dually, the number of predators y decreases in proportion to the number of predators
either because they starve off or emigrate. This exponential decay is corrected by the
encounters of predators with preys. This can be modeled mathematically with the
following differential equation system with α, β, γ , δ > 0 (Figure 2.1).
dx
= αx − βx y ,
dt
dy
= −γ y + δx y .
dt

(2.5)

Deterministic versus stochastic models: In a deterministic model no randomness is
involved in the development of future states of the modeled system. In other words,
a deterministic model will always produce the same output starting with a given set
of initial conditions as in the example above. A stochastic model, on the other hand,
includes randomness. The states of the modeled system do not have unique values.

future behavior of large groups of populations—like the Galactic Empire with a quintillion citizens

[7]?

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2.2 What Is a Model?

11

They must rather be described by probability distribution functions. A good example in physics is the movement of small particles in a liquid—Brownian motion—
resulting from their bombardment by a vast number of fast moving molecules [36].
Finance uses mostly stochastic models for predictions. These are discussed in detail
in Chapter 5.
Random or Chaotic?
Webster Dictionary defines “random” as “a haphazard course; without definite aim,
direction, rule, or method”. The same source defines “chaos” as “complete confusion
and disorder; a state in which behavior and events are not controlled by anything”.
Those descriptions sound quite similar. However, in the mathematical or engineering
context the word “chaos” has a very specific meaning. Chaos theory is a field in mathematics which studies the behavior of dynamic systems that are extremely sensitive to
initial conditions. Tiny differences in initial conditions result in completely different
outcomes for such dynamic systems. As the frequently told anecdote goes, Edward
Norton Lorenz, a meteorologist with a strong mathematical background in non-linear
systems, was making weather predictions using a simplified mathematical model for
fluid convection back in 1961. The outcomes of the simulations were wildly different
depending on whether he used three digits or six digits to enter the initial conditions
of the simulations. Although it was known for a long time that non-linear systems
had erratic or unpredictable behavior, this experience of Lorenz became a monumental
reminder that even though these systems are deterministic, meaning that their future
behavior is fully determined by their initial conditions, with absolutely no random elements involved, they cannot be used to predict the future for any meaningful purpose.
Lorenz is reputed to have quipped “Chaos is when the present determines the future,

but the approximate present does not approximately determine the future”.
A double pendulum is made of two rods attached to each other with a joint. One of
the pendulums is again attached with a joint to a base such that it can revolve freely
around that base. Anyone who watches this contraption swing for some time cannot
help being mesmerized by its unpredictable slowing downs and speeding ups. The
double pendulum can be modeled very accurately by a non-linear ordinary differential
equation (see, for instance, [135] for a complete derivation). However, when one tries to
simulate (or solve numerically) this equation one will observe the sensitive dependency
of its behavior on the initial conditions one chooses.
There is even a measure for chaos. The Lyapunov exponent of a dynamic system is a
quantity that characterizes the rate of separation of infinitesimally close trajectories.
Two trajectories in state space with initial separation δx0 diverge at a rate given by
|δx(t)| ≈ eλt |δx0 | where λ denotes the Lyapunov exponent.

Static versus dynamic models: In static models, as the name suggests, the effect of time is
not considered. Dynamic models specifically account for time. Such models make use of
difference or differential equations with time as a free variable. Looking at the input-output
relationship of an amplifier far from saturation, a static model will simply consist of the
amplification factor. A dynamic model, on the other hand, will use a time function to describe
how the output changes dynamically, including its transient behavior, as a consequence of
changes in the input [26].


12

2 Modeling and Identification

Discrete versus continuous models: Some recent models of quantum gravity [122] notwithstanding, in the current Weltbild time flows smoothly and continuously. Models building on
that—often tacit—assumption make use of differential equations, solutions of which are time
continuous functions. However, the advent of the digital computer which works with a clock

and spews out results at discrete time points made it necessary to work with discrete time models which are best described using difference equations. Whereas the classical speed control
by means of a fly ball governor is based upon a continuous time model of a steam engine [130],
modern robot movement controllers employing digital processors use discrete time models
[106].
Lumped parameter models versus distributed parameter models: Distributed parameter or
distributed element models assume that the attributes of the modeled system are distributed
continuously throughout the system. This is in contrast to lumped parameter or lumped element models, which assume that these values are lumped into discrete elements. One example
of a distributed parameter model is the transmission line model which begins by looking at the
electrical properties of an infinitesimal length of a transmission line and results in the telegrapher’s equations. These are partial differential equations involving partial derivatives with
respect to both space and time variables [92]. Equations describing the elementary segment
of a lossy transmission line developed by Heaviside5 in 1880 as shown in Figure 2.2 are
∂
∂
V (x, t) = −L I (x, t) − R I (x, t) ,
∂x
∂t
∂
∂
I (x, t) = −C V (x, t) − GV (x, t) .
∂x
∂t

(2.6)
(2.7)

V (x, t) and I (x, t) denote the position and time dependent voltage and current respectively. The parameters R, G, L , C are the distributed resistance, conductance, inductance and
capacitance per unit length. A lumped parameter model simplifies the description of the behavior of spatially distributed physical systems into a topology consisting of discrete entities that
approximate the behavior of the distributed system. The mathematical tools required to analyze such models are ordinary differential equations involving derivatives with respect to the
time variable alone. A simple example for a lumped parameter models is an electrical circuit
consisting of a resistor and an inductor in series. Such a circuit can adequately be described

by the ordinary differential equation

Fig. 2.2 Schematic
representation of the
elementary components of a
transmission line with the
infinitesimal length d x

5 Oliver

Heaviside, English mathematician and engineer (1850–1925).

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